On the Sets of Stability to Perturbations of Some Continued Fraction with Applications

This paper investigates the stability of continued fractions with complex partial denominators and numerators equal to one. Such fractions are an important tool for function approximation and have wide application in physics, engineering, and mathematics. A formula is derived for the relative error of the approximant of a continued fraction, which depends on both the relative errors of the fraction’s elements and the elements themselves. Based on this formula, using the methodology of element sets and their corresponding value sets, conditions are established under which the approximants of continued fractions are stable to perturbations of their elements. Stability sets are constructed, which are sets of admissible values for the fraction’s elements that guarantee bounded errors in the approximants. For each of these sets, an estimate of the relative error that arises from the perturbation of the continued fraction’s elements is obtained. The results are illustrated with an example of a continued fraction that is an expansion of the ratio of Bessel functions of the first kind. A numerical experiment is conducted, comparing two methods for calculating the approximants of a continued fraction: the backward and forward algorithms. The computational stability of the backward algorithm is demonstrated, which corresponds to the theoretical research results. The errors in calculating approximants with this algorithm are close to the unit round-off, regardless of the order of approximation, which demonstrates the advantages of continued fractions in high-precision computation tasks. Another example is a comparative analysis of the accuracy and stability to perturbations of second-order polynomial model and so-called second-order continued fraction model in the problem of wood drying modeling. Experimental studies have shown that the continued fraction model shows better results both in terms of approximation accuracy and stability to perturbations, which makes it more suitable for modeling processes with pronounced asymptotic behavior.